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Full text of "BSTJ : Pressure at the Interface of Inner Conductor and Dielectric of Armorless Ocean Cable (Bhuta, P.G.)"

Pressure at the Interface of Inner 

Conductor and Dielectric of 

Armorless Ocean Cable 

By P. G. BHUTA 

(Manuscript received February 8, 1960) 

Pressure at the interface of the inner conductor and the dielectric of the 

armorless ocean cable is experimentally determined for -plane stress boundary 
conditions. Polyethylene older and steel inner concentric cylinders are con- 
sidered as a mechanical model of the ocean cable. Equations arc derived, to 
give the pressure at the interface for plane stress and plane strain conditions. 
It is found that the theoretical values of the pressure calculated from equa- 
tions of elasticity are in good agreem.ent mith experimental values, even when 
one of the materials is polyethylene, which has a nonlinear stress-strain 
relationship in simple tension or compression. From previously measured 
values of bulk and Young's moduli, it is shown that, for an elastic analysis, 
Poisson's ratio for polyethylene is very nearly 0.5. An experimental verifica- 
tion of the plane stress solution given in this paper confi.rms the foregoing 
value. 

I. INTRODUCTIOX 

In the process of designing new armorless coaxial ocean cables it is 
important to know the pressure at the interface of the inner conductor 
and the polyethylene dielectric. Dotermuiation of pressure at the inter- 
face would enable one to study the decay of tension, hi a cable-laying 
machine, from the strength member to the holding surface, and to in- 
vestigate the pressure effects on the resistance and capacitance of the 
cable. 

A suitable mechanical model of the armorless ocean cable is a poly- 
ethylene outer cylinder concentric with a steel inner cylinder. The caljle 
is actually subjected, m three dimensions, to a pressure loading that 
vari'-s with the depth of the ocean bottom. Ilowever, for a section of the 
cable on a relatively flat ocean l)ottom one may assume that the cable 
is subjected to a uniform external pressure. 

963 



964 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 

One may obtain an expression for the pressure at the interface from a 
two-dimensional model (i.e., for plane stress or plane strain) using the 
theory of elasticity. However, the result is questionable, since poly- 
ethylene has a nonlinear stress-strain relationship in simple tension or 
compression. Moreover the value of Poisson's ratio for polyethylene, 
fip , ranges from 0.3 to 0.5, and it is not possible to determine iXp accu- 
rately by a simple experiment. The value of the pressure at the interface, 
as obtained from such an equation, depends considerably upon the 
particular value of fip that is used. Hence, an experimental determination 
of the pressure at the interface is necessary, to investigate whether an 
elastic analysis is valid when one of the materials is polyethylene and to 
establish the value of Poisson's ratio for polyethylene. 

The pressure at the interface of the model is determined experimentally 
for plane stress boundary conditions. Equations are derived, based on 
the two-dimensional theory of elasticity, to give the pressure at the in- 
terface of the model for plane stress and plane strain conditions. It is 
found that the theoretical values of the pressure calculated from equa- 
tions of elasticity are in good agreement with experimental vahies. From 
previously measured values of bulk and Young's moduli, it is shown that, 
for an elastic analysis, Poisson's ratio for polyethylene is very nearly 0.5. 
For this reason and since the area of polyethylene is much greater than 
that of steel, an extension of the two-dimensional plane strain solution 
to a three-dimensional case of uniform pressure by the method of super- 
position would give the pressure at the inner conductor and dielectric 
interface to be equal, or at most slightly greater, than the external pres- 
sure. 

II. EXPERIMENTAL MODEL AND PROCEDURE 

Initially, the experunental model consisted of a cold-drawn steel tub- 
ing 30 inches long, 1.254 inches in outer diameter and 0.083 inch in wall 
thickness. Two SR-4 strain gages of type A7 were cemented on a O.OIO- 
inch-thick steel ring, which fitted snugly inside the steel tubing shown 
in Fig. 1. The steel ring containing the gages to which long leader wires 
were soldered was cemented in the tubing so that the it was at the same 
distance from either end. The gages were far enough from the ends to 
eliminate end effects. 

The tubing was subjected to an external hydrostatic pressure in in- 
crements of 500 psi up to 4500 psi when a maximum strain of approxi- 
mately 1000 microinches per inch, which is well within the elastic limit 
for the tubing, was recorded. The pressure was reduced to zero, and upon 
increasing the pressure again, the same calibration curve was obtained. 



I-RESSURE AT INTERFACE IX ARMORLESS OCEAN CABLE 



OHo 



■30 



i.a54 



.it" 



SEE DETAIL A — 



GAGE 1 

GAGE 2 _ 



"-^m 



tZZZZZZ222ZZZZl 



0.083 



1 " 
-^ 2 [^ 0.010' 



Fig. 1 — Steel tubing, showing location of strain gages. 

The experimental arrangement for calibration of gages is sho^^ni in 

Fig. 2. 

After calibration of gages, the polyethylene outer cylmder, whose in- 
ner diameter was 0.004 inch smaller than the outer diameter of the steel 
cylinder, was forced onto the steel cylinder in a universal testing ma- 
chine. The assembled model appeared as in Fig. 3. Before the model was 
subjected to external hydrostatic pressure, the strain gages were bal- 
anced to indicate zero strain by adjusting resistances in a multichannel 
switching-and-balancing unit. The experimental arrangement for the 
model is given in Fig. 4. It is not the intent of the present paper to evalu- 
ate the pressure produced at the interface by the interference fit. External 
hydrostatic pressure was applied in increments of 500 psi up to 2500 psi 
and corresponding strains were observed. The pressure was reduced to 
zero and the reproducibility of the data was checked. 







^ ^"O" RING 
I / SEAL 



GAGE l--^--GAGE 2 

ITTTTTTmTTTni 



^^m^/^^^^^^^Z^^^^ 




'-STEEL 



Fig. 2 — Experimental arrangement for calibration. 



9r»6 



THE BKLL SYSTEM TECHNICAL JOURNAL, JULY 1960 




Fig. 3 — Experimental model. 




-POLYETHYLENE 



Fig. 4 — Experimental arrangement for the model. 
III. EXPERIMENTAL RESULTS 

The calibration curves for external hydrostatic pressure versus the 
strain observed froin gages 1 and 2, when the external hydi-ostatic pres- 
sure was applied directly on the pipe, are given in Figs. 5 and 6. Corre- 
spondmg curves when the external pressure was applied on the model 
are also plotted in these figures. 

The lines pa.ss through the origin because the bridge was initially 
balanced at zero pressure with the help of blaiicing resistances in the 
multichannel switching unit, and in each case the zero reading was 
checked after the experiment, when the pressure was brought back to 
zero. The deviation of the experimental points from the straight hnes for 
low values of pressure is probably due to the nonlinearity in the pressure 
gages, which are designed to be used normally at much higher pressures. 

It may be noted that, in the case of Lame's equations for thick-walled 
cylinders, the sum of the radial and circumferential stresses is a con- 
stant at every point in the cross section, so that the deformation of all 
elements in the direction of the axis of the cylmder is the same and cross 
sections of the cylinder remain plane after deformation. Hence, in spite 
of the considerable length of the cyhnder, the experunental arrangement 
gives the verification for plane stress conditions. 



PRESSURE AT IMTERFACE IX ARMORLESS OCEAN CABLE 



967 



5000 























f 




O 

z 


















1^ 


/ 






u^OOO 
cc 

= 3500 
10 










CALIBRATION FOR 
STEEL CYLINDER^ 


/ 




Z' 
















/ 


/ 


y 


y 






0. 












y 


/ 


y 


^ 








z 
3 










/ 


/ 
^ 


y 












z 








/ 


r 
/ 


/concentric STEEL AND 
POLYETHYLENE CYLINDERS 




PRESSURE 

8 i 






4' 


y 


^T. 
















/ 


/y 


y 























y 























400 600 eoo 

STRAIN, INCH PER INCH 



1000 I2O0XI0'' 



Fig. 5 — Pressure vs, strain from giige 1. 

The pressure iit the interface is obtained hy druwinfj; a vertical Hne 
from the observed vahie of strain to the caliliration line for the steel 
cylinder alone and reading the eorrespondmg pressure. Numerical calcu- 
lations from Figs. 5 and 6 show that the pressure at the interface, in the 
case of plane stress is 1 .3 times the external pressure. 



5000 


























z 




















/ 


/ 




a. 

g3500 
in 










CALIBRATION FOR 
STEEL CYLINDER 


/ 






















/I 




/ 


¥^ 




Q. 














/ 


-£y 


/ 








z 

D 

2 2000 

z 

UJ 1500 
a: 










./ 


V 


y 


1 














/ 


/ 


/concentric steel AND 
'^ POLYETHYLENE CYLINDERS 








/ 


^ 


•■^ 
















0. 


9 


/- 


r 























i^ 














i 









200 



400 600 800 

STRAIN, INCH PER INCH 



1200 X 10^ 



Fig. — Pressure vs. strain from gage 2. 



968 THE BICLL SYSTEM TECHNICAL JOURNAL, JULY 1960 



,-- — STEEL 




- — POLYETHYLENE 



Fig.. 7 — Cross section of the model. 
IV. THEORETICAL DERIVATION OF PRESSURE AT THE INTERFACE 

The differential cqimtion of equilibrium,^ in terms of radial displace- 
ment, for a thick-walled cylinder subjected to external and intermal 
pressures is 



du 1 du ^* _ r, 
dr"^ r dr t' 



The general solution of the above equation is 



(1) 



r 



(2) 



Substituting (2) in the expressions for Hooke's law fur plane stress, one 
obtams for radial stress, a, 



1 -m4 



Cl(l + ;U) — C2 



(1 - 



1 -m) 1 
r= J' 



(3) 



where Ci and Ca are arbitrary constants and E and m are Young's modulus 
and Poisson's ratio respectively. For the model shown in Fig. 7, the 
following boundary conditions must be satisfied; 






(4) 
(5) 



PKESSURE AT INTERFACE IN ARMORLESS OCEAN CABLE 969 

C,\r=.= 0, (6) 



Up r=6 ^ Ws T=b • 



(7) 



In the above equations subscripts p and s stand for polyethylene and 
steel. Substituting (2) and (3) in (4) through (7), one gets: 



^[.(i + .)-.(Lz^')] 



^-A^[.a + .)-c.(L^-)]. 



K r 
1 - Mp^ L 



Ci(l + Up) — G2 



(1 - 



1 - Mp) 1 
C2 J 



= ?Jo, 



,; Cad + mJ - Ci -T. — ■ = 0, 



Cih -h ~ = Czb + r* . 



(8) 

(9) 
(10) 
(11) 



The pressure p at the interface is obtained by solving for the constants 
from the foregoing equations and evaluating 

(Tp I r=b = p. 

Equation (12) gives the pressure at the interface for a plane stress 
solution. 



P = VoS 



(b' 


-c)\^(l -,.p)(h' -a') 




- (1 - M.)&' - (1 + M.)«' 


(6^ 


-c')[(l - mJ&' + (1 + M.)a1 




-^(b' - a') [(1 - ^p)b' + (1 + ^ip)c] 
hp 



+ 



. (12) 



To obtain the plane strain solution, the constants E and n in (12) are 
replaced by E* and m*, with appropriate subscripts given by the follow- 



ing; 



E* = 



M* = 



B 



It , 

1 -m' 



(13) 

(14) 



970 THE BELL SYSTEM TECHNICAL JOUHNAL, JULY ] 9G0 

V. NUMERICAL EXAMPLE FOB THE MODEL 

The values for the various constants are : 

a = 1.088 inches, 

h — 1.254 inches, 

c = 4.621 inches, 

i:. - 29 X 10' psi, 

7?^ = 19 X 10' psi, 

Us = 0.3, 

M;, = 0.5. 

Substitution of above values in (12) gives for tlie plane stress solution 

V= 1.3(po). (15) 

Replacing E and n by (13) and (14) and substituting in (12), one 
obtains for the plane strain solution 

V = l-0(p«). (16) 

It may be noticed from (12) that the pressure at the interface ap- 
pears to depend on the values of Ep and Hp . However, examination 
reveals that (12) is relatively msensitive to Ep , for the range of known 
values for low and intermediate density polyethylene. Since it is difficult 
to determine tip , one may use the previously known values of bulk 
modulus, Kj, , for polyethylene and obtaui Hp from 

Refs. 3 and 4 give the value of Kp ^ 0.5 X 10^ psi. Ref. 5 gives for E,, , 
for low and intermediate density, the range of values from 19 X 10^ to 
55 X 10' psi. With Kp = 0.5 X lO' psi and Ep - 19 X lO' or 55 X lO' 
psi, (17) gives Mp to be very nearly equal to 0.5. It may be noted that 
the analysis for concentric cylinders given in this paper agrees with 
experimental results when /Zp is very nearly equal to 0.5. 

VI. CONCLUSION 

Experiments indicate that an analysis based on equations of the two- 
dimensional theory of elasticity for the pressure at the mterface of 
polyethylene outer and steel inner concentric cylinders, under plane 



PRESSURE AT INTERFACE IN ARMORLESS OCEAN CABLE 071 

stress conditions subjected to a uniform pressure on the curved surface, 
is valid even for polyethylene, which is a material with nonlinear stress- 
strain relationship. The agreement between the theoretically calculated 
and experimentally determined values is very good. It is also shown 
that the value of Poisson's ratio for low and intermediate density poly- 
ethylene, for an elastic analy.sis, should be very nearly 0.5. The experi- 
mental verification of the plane stress .solution given in the present paper 
confirms this value. 

An experimental verification of the plane strain solution involves a 
number of difficulties in satisfying the condition that the axial strain be 
zero; it has not been attempted in the present paper. However, one may 
assume that the plane strain solution is also valid, because the plane 
stress solution obtained by using the same two-dimensional theory of 
elasticity has been experimentally verified. The plane strain solution 
gives for the numerical example the result that the pressure at the inter- 
face is the same as external pressure. 

The cable is actually under a uniform pressure in three dimensions. 
The two-dimensional plane strain solution may be extended to a three- 
dimensional case of uniform pressure by a superposition of an axial 
loading of intensity po and satisfying the appropriate boundary and 
compatibility conditions. However, since the value of Poisson's ratio for 
polyethylene is 0.5 and the area of polyethylene is much greater in 
comparison to the area of the steel, the pre.ssure at the interface of the 
inner conductor and the polyethylene dielectric, as given by the plane 
strain solution, would not be much altered. Hence, the pressure at the 
interface of the mner conductor and the dielectric of the armorless ocean 
cable is equal to or at most slightly greater than the external pressure. 

It may be remarked that a new experimental method employing re- 
sistance wire strain gages has been developed to measure interface pres- 
sures accurately. The method devised in this paper could be used in 
other applications to mea.sure interface pres.sures. 

VII. LIST OF SYMBOLS 

a = inner radius of steel cylinder (inches). 
h = outer radius of steel cylinder (inches), 
c = outer radius of polyethylene cylinder (inches). 
Kp = bulk modulus for polyethylene (psi). 
p = pres.sure at the interface (psi). 
^,(1 = external hydrostatic pres^;urc (psi). 
u ^ radial displacement (inches). 
r = radial distance (inches). 



972 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1900 

E = Young's modulus (psi). 
Ep = Young's modulus for polyethylene (psi). 
E, = Young's modulus for steel (psi). 

n = Poisson's ratio. 
fip = Poisson's ratio for polyethylene. 
fig = Poisson's ratio for steel. 

a = radial stress (psi). 
tTp = radial stress in polyethylene (psi). 
a, = radial stress m steel (psi). 

REFERENCES 

1. Timoflhenko, S., Strength of Materials, Part II, D. Van Nostrand Co., New 

York, 1956, pp. 205-214. 

2. Timoshenko, S. and Goodier, J. N., Theory of Elasticity, McGraw-Hill Book 

Co., New York, 1951, pp. 34; 54-60. 

3. Bridgman, P. W., Proc. Amer. Acad. Arts & Sei., 76, 1948, p. 71. 

4. Weir, C. E., National Bureau of Standards, Research Paper 2192, 1951. 

5. Modern Plastics Encyclopedia, Plastics Properties Chart, Breskin Publications, 

New York, 1959.